Abstract
A new analytical and numerical solution of the electrodynamic waveguide problem for leaky modes of a planar dielectric symmetric waveguide is proposed. The conditions of leaky modes, corresponding to the Gamow-Siegert model, were used as asymptotic boundary conditions. The resulting initial-boundary problem allows the separation of variables. The emerging problem of the eigen-modes of open three-layer waveguides is formulated as the Sturm-Liouville problem with the corresponding boundary and asymptotic conditions. In the case of guided and radiation modes, the Sturm-Liouville problem is self-adjoint and the corresponding eigenvalues are real quantities for dielectric media. The search for eigenvalues and eigenfunctions corresponding to the leaky modes involves a number of difficulties: the problem for leaky modes is not self-adjoint, so the eigenvalues are complex quantities. The problem of finding eigenvalues and eigenfunctions is associated with finding the complex roots of the nonlinear dispersion equation. To solve this problem, we used the method of minimizing the zero order. An analysis of the calculated distributions of the electric field strength of the first three leaky modes is given, showing the possibilities and advantages of our approach to the study of leaky modes.
Highlights
In the books by Marcuse [1], [2], Adams [3], Snyder and Love [4], Tamir [5], and other authors the terms “leaky rays” and “leaky modes” appear when discussing the propagation of polarized light in fiber optical waveguides with the refractive index of the core smaller than that of the cladding, and in planar waveguides with plates of material optically denser than the waveguide layer itself
A more rigorous justification of the model of leaky waves of open waveguide systems can be obtained by starting calculations not from the Helmholtz equation, as is traditionally done, but from the wave equation preceding the Helmholtz equation, and most importantly, more adequately reflecting the wave nature of leaky modes
In the case of a field corresponding to leaky modes running in the positive direction of the z-axis for x > h and x > h due to the symmetry of the waveguide, the wave vector is determined as k⃗j± = k0 ⎛⎜±√n2c − βj2⎞⎟
Summary
In the books by Marcuse [1], [2], Adams [3], Snyder and Love [4], Tamir [5], and other authors the terms “leaky rays” and “leaky modes” appear when discussing the propagation of polarized light in fiber optical waveguides with the refractive index of the core smaller than that of the cladding, and in planar waveguides with plates of material optically denser than the waveguide layer itself. As shown by Marcuvitz [23], these complex poles can correspond to leaky modes They do not make a direct contribution to the correct spectral solution and can be characterized by non-physical growth towards infinity, they can accurately describe the radiation field in limited spatial domains. There is an urgent need to develop new algorithms for calculating the fields of both radiative and leaky modes, surpassing the standard methods, e.g., the FDTD method, in count rate and not inferior to them in accuracy In quantum physics, such solutions of the stationary Schrödinger equation are called Gamow resonances [36], [37] or Siegert quasi-states [38]. A more rigorous justification of the model of leaky waves of open waveguide systems can be obtained by starting calculations not from the Helmholtz equation, as is traditionally done, but from the wave equation preceding the Helmholtz equation, and most importantly, more adequately reflecting the wave nature of leaky modes
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